ALGEBRAS, SPACES, LOGICS. 519 



case. But let me pause here. "We have sufficiently shown our plural 

 without even mentioning Cayley and Sylvester's invariantive algebra ; 

 Riemann's theory of a complex variable ; the algebra of polar elements ; 

 or any of the many others that have sprung or are springing into 

 being. 



As for pluralizing the idea of space, that would follow very briefly 

 if only I might talk in terms of the " Ausdehnungslehre." Quaternions, 

 as Professor Tait has said, is content with one flat space ; but Grass- 

 raann, in a little appendix of only two pages, has shown the ability of 

 his extensive algebra to cope with the modern double plural of the old 

 idea of space. Before this idea had germinated, while therefore there 

 was no real use for the word "spaces," the parsimony of language 

 applied it to mean jeces of space ; but in the fullness of time it has 

 received its heritage, and by spaces I mean an aggregate of which the 

 space hypothetically infinite and containing the material universe is 

 but one. A statement in the technical terms of analysis would prob- 

 ably tend very little toward clearing up this matter to one not already 

 familiar with it. Let us, then, use rather the historical method attack 

 in the light of history. 



As an eternal treasure and model to the world the Greeks be- 

 queathed the synthetic science of a space. This is the particular 

 space in which you believe, and are sure you and the stars are inhab- 

 iting. You will be glad to know that it has been made a fitting monu- 

 ment to the writer of the greatest classic, and inscribed with the name 

 of Euclid. This Euclidean space is a tridimensional homaloid, and so, 

 in distinction from it, spaces with positive or negative curvature are 

 called non-Euclidean. 



Through all the centuries up to the present Euclid's space contained 

 at least the thought-world. The space analyzed in Euclid's " Elements " 

 was supposed to be the only possible form, the only non-contradictory 

 sort of space. And, after more than twenty centuries, it is to a little 

 point in that same book that the new idea attaches itself and sprouts 

 into being. This slender link is one of Euclid's postulates, misplaced 

 in the English editions as the twelfth axiom. As the last of his six 

 alT?]aara (requests) Euclid says : " Let it be granted that if a straight 

 line meet two other straight lines, so as to make the two interior angles 

 on the same side of it, taken together, less than two right angles, 

 these straight lines being continually produced shall at length meet 

 upon that side on which are the angles, which are together less than 

 two right angles." This somewhat complicated so-called axiom is only 

 the converse or inverse of proposition seventeen, that " any two angles 

 of a triangle are together less than two right angles," a theorem readily 

 demonstrated from the preceding postulates and axioms. An inverse is 

 usually exceedingly easy to prove. Then why not remove this inverse 

 from among the postulates, place it after seventeen, and demonstrate 

 it ? This obvious way to improve on Euclid suggested itself to nu- 



